• Climate change is well understood globally.

• Climate change is less well understood locally.

• Need for spatailly explicit reconstructions of climate variables.

• Problem: data souruces are messy and noisy.

• Many ecological and phyiscal processes respond to climate over different time scales.
  • Tree rings, corals, forest landscapes, ice rings, lake levels, etc.

• These processes are called climate proxies.
  • They are proxy measurements for unobserved climate.
  • Noisy and messy.
  • Respond to a wide variety of non-climatic signals.

• Bayesian hierarchical model.

\(\begin{align*} [\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}] & \propto [\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}] [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)

• Bayesian hierarchical model.

\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto [\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}] [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)

• Posterior.

• Bayesian hierarchical model.

\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)

• Posterior.

• Data Model

• Bayesian hierarchical model.

\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} \color{blue}{[\mathbf{Z} | \boldsymbol{\theta}_P]} [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)

• Posterior.

• Data Model.

• Process Model.

• Bayesian hierarchical model.

\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} \color{blue}{[\mathbf{Z} | \boldsymbol{\theta}_P]} \color{orange}{[\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P]} \end{align*}\)

• Posterior.

• Data Model.

• Process Model.

• Prior Model.

• Describes how the data are collected and observed.

• Researchers take sediment samples from a lake.

• Take 1cm\(^3\) cubes along the length of the sediment core.

• In each cube, researcher counts the first \(N\) pollen grains and identifies to species.

• Raw data are counts of each species

For location \(\mathbf{s}\) and time \(t\),

\(\begin{align*} \mathbf{y} \left( \mathbf{s}_i, t \right) & = \left( y_{1} \left( \mathbf{s}_i, t \right), \ldots, y_{d} \left( \mathbf{s}_i, t \right) \right)' \end{align*}\)

is an observation of a \(d\)-dimensional compositional count.

• \(y_{j} \left( \mathbf{s}_i, t \right)\) is the count of species \(j\) in the sample at location \(\mathbf{s}_i\) and time \(t\).

• Compositional count data.
  • Total count is not informative of the absolute composition.
  • Informative of the relative proportions \(p_{j} \left( \mathbf{s}_i, t \right)\) only.

• Compositional count vector \(\mathbf{y} \left( \mathbf{s}_i, t \right)\) a function of latent proportions \(\mathbf{p}\left( \mathbf{s}_i, t \right)\)

\(\begin{align*} \mathbf{y}\left( \mathbf{s}_i, t \right) | \mathbf{p}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Multinomial} \left( N\left( \mathbf{s}_i, t \right), \mathbf{p}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)

• \(N\left( \mathbf{s}_i, t \right) = \sum_{j=1}^d y_{j}\left( \mathbf{s}_i, t \right)\) is the total count observed (fixed and known) for observation at location \(\mathbf{s}_i\) and time \(t\).

• Compositional count vector \(\mathbf{y} \left( \mathbf{s}_i, t \right)\) a function of latent proportions \(\mathbf{p}\left( \mathbf{s}_i, t \right)\)

• The pollen data are highly variable and overdispersed.

\(\begin{align*} \mathbf{p}\left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Dirichlet} \left( \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)

• Marginalize out \(\mathbf{p} \left( \mathbf{s}_i, t \right)\) to get Dirichlet-multinomial

\(\begin{align*} \mathbf{y}\left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Dirichlet-Multinomial} \left( N\left( \mathbf{s}_i, t \right), \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)

• We model the Dirichlet-multinomial count data using the log link function:

• \(\begin{align*} \operatorname{log} \left( \boldsymbol{\alpha} \left( \mathbf{s}_i, t \right) \right) & = \mathbf{z}\left( \mathbf{s}_i, t \right) \boldsymbol{\beta}. \end{align*}\)

• \(\mathbf{z}\left( \mathbf{s}_i, t \right)\)’ is a \(q\)-dimensional vector of covariates.

• \(\boldsymbol{\beta}\) is a \(q \times d\) dimensional matrix of regression coefficients.

• The \(\mathbf{z} \left( \mathbf{s}_i, t \right)\)s are observed only at \(t\) = 1.

• Calibration:
  • Estimate \(\boldsymbol{\beta}\) using:
    • \(\left( \mathbf{y} \left( \mathbf{s}_1, 1 \right), \ldots, \mathbf{y} \left( \mathbf{s}_n, 1 \right) \right)'\) and
    • \(\left( \mathbf{z} \left( \mathbf{s}_1, 1 \right), \ldots, \mathbf{z} \left( \mathbf{s}_n, 1 \right) \right)'\).

• Reconstruction:
  • Use estimated \(\boldsymbol{\beta}\)s and fossil pollen \(\mathbf{y} \left( \mathbf{s}, t \right)\) to predict unobserved \(\mathbf{z}\left( \mathbf{s}, t \right)\).

• Vegetation response to climate is non-linear.

• \(\begin{align*} \operatorname{log} \left( \boldsymbol{\alpha} \left( \mathbf{s}_i, t \right) \right) & = \mathbf{B} \left( \mathbf{z}\left( \mathbf{s}_i, t \right) \right) \boldsymbol{\beta} \end{align*}\)

• \(\mathbf{B} \left( \mathbf{z}\left( \mathbf{s}_i, t \right) \right)\) is a basis expansion of the covariates \(\mathbf{z}\left( \mathbf{s}_i, t \right)\).
  • Use B-splines or Gaussian Processes as a basis.

• Note that for \(t \neq 1\), the \(\mathbf{z} \left( \mathbf{s}_i, t \right)\)s are unobserved

• We are interested in estimating the latent process \(\mathbf{z} \left( \mathbf{s}, t \right)\).

• The model can accommodate:
  1. continuous vs. discrete space (geostatistical vs. CAR models)
  2. continuous vs. discrete time (stochastic process vs. AR models)

• For now, we focus on continuous space and discrete time

• For \(\mathbf{z} \left(t \right) = \left( \mathbf{z} \left(\mathbf{s}_1, t \right)', \ldots, \mathbf{z} \left(\mathbf{s}_n, t \right)' \right)\), we assume:

\(\begin{align*} \mathbf{z} \left(t \right) - \mathbf{X} \left( t \right) \boldsymbol{\gamma} & = \mathbf{M}\left(t\right) \left( \mathbf{z} \left(t-1 \right) - \mathbf{X} \left( t \right) \boldsymbol{\gamma} \right) + \boldsymbol{\eta} \left(t \right) \end{align*}\)

• \(\mathbf{M}(t) = \rho \mathbf{I}_n\) is a propogator matrix

• \(\mathbf{X} \left(t \right) \boldsymbol{\gamma}\) are the fixed effects from covariates like latitude, elevation, etc.

• \(\boldsymbol{\eta} \left( t \right) \sim \operatorname{N} \left( \mathbf{0}, \tau^2 \mathbf{R}\left( \boldsymbol{\phi} \right) \right)\)

• \(\tau^2\) is the spatial process variance

• \(\mathbf{R} \left( \boldsymbol{\phi} \right)\) is a Mátern spatial covariance matrix with parameters \(\boldsymbol{\phi}\)

• Each fossil pollen observation includes estimates of time uncertainty.
  • The time of the observation is uncertain.
  • Weight the likelihoods according to age-depth model.
  • Posterior distribution of ages.

• For each observation fossil pollen observation an age-depth model gives a posterior distribution over dates.
  • Define \(\omega \left(\mathbf{s}_i, t \right)\) as P(age \(\in (t-1, t)\)). Then
  • \([\mathbf{y} \left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha} \left( \mathbf{s}_i, t \right) ] = \prod_{t=1}^T [\mathbf{y} \left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha} \left( \mathbf{s}, t \right)]^{\omega_\left(\mathbf{s}_i, t \right)}\)